Systems of Parameters and the Cohen-Macaulay Property

Let R be a commutative, Noetherian, local ring and M a finitely generated R-module. Consider the module of homomorphisms HomR(R/ a,M/bM) where b ⊆ a are parameter ideals of M. When M = R and R is Cohen-Macaulay, Rees showed that this module of homomorphisms is isomorphic to R/ a, and in particular, a free module over R/ a of rank one. In this work, we study the structure of such modules of homomorphisms for a not necessarily Cohen-Macaulay R-module M